Numerical Approximations for the Tempered Fractional Laplacian: Error Analysis and Applications

Duo, Siwei; Zhang, Yanzhi

doi:10.1007/s10915-019-01029-7

Cited by 24 publications

(19 citation statements)

References 23 publications

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“…This operator was used to develop finite difference schemes to solve the tempered fractional Laplacian equation that governs the probability distribution function of the positions of particles. Similarly, Duo et al [31] presented a finite difference method to discretize the d-dimensional (for d ≥ 1) tempered integral fractional Laplacian (−∆ + λ) α/2 . By means of this approximation they resolved fractional Poisson problems.…”

Section: Introductionmentioning

confidence: 99%

Bilateral Tempered Fractional Derivatives

Ortigueira

Bengochea

2021

Symmetry

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Section: Introductionmentioning

confidence: 99%

Bilateral Tempered Fractional Derivatives

Ortigueira

Bengochea

2021

Symmetry

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“…This operator was used to develop finite difference schemes to solve the tempered fractional Laplacian equation that governs the probability distribution function of the positions of particles. Similarly, Duo et al [30] presented a finite difference method to discretize the d-dimensional (for d ≥ 1) tempered integral fractional Laplacian (−∆ + λ) α/2 . By means of this approximation they resolved fractional Poisson problems.…”

Section: Introductionmentioning

confidence: 99%

Bilateral Tempered Fractional Derivatives

Ortigueira¹,

Bengochea²

2021

Preprint

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“…On the other hand, the integral definition in (3) provides a pointwise formulation and thus can easily work with different boundary conditions. Based on the integral definition, finite difference methods [15,10,14] and finite element methods [2,1,7,4] have been developed to discretize the fractional Laplacian (−∆) α 2 with extended Dirichlet boundary conditions. In contrast to the pseudo-differential definition, the integral definition in (3) is valid only for α ∈ (0, 2), so do the resulting numerical methods [15,10,14,2,1,7,4].…”

mentioning

confidence: 99%

“…Based on the integral definition, finite difference methods [15,10,14] and finite element methods [2,1,7,4] have been developed to discretize the fractional Laplacian (−∆) α 2 with extended Dirichlet boundary conditions. In contrast to the pseudo-differential definition, the integral definition in (3) is valid only for α ∈ (0, 2), so do the resulting numerical methods [15,10,14,2,1,7,4]. Moreover, the above methods based on the integral definition are mainly limited to the second order accuracy.…”

mentioning

confidence: 99%

Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization

Zhang

2022

DCDS-S

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<p style='text-indent:20px;'>In this paper, we propose a new class of operator factorization methods to discretize the integral fractional Laplacian <inline-formula><tex-math id="M1">\begin{document}$ (- \Delta)^\frac{{ \alpha}}{{2}} $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M2">\begin{document}$ \alpha \in (0, 2) $\end{document}</tex-math></inline-formula>. One main advantage is that our method can easily increase numerical accuracy by using high-degree Lagrange basis functions, but remain its scheme structure and computer implementation unchanged. Moreover, it results in a symmetric (multilevel) Toeplitz differentiation matrix, enabling efficient computation via the fast Fourier transforms. If constant or linear basis functions are used, our method has an accuracy of <inline-formula><tex-math id="M3">\begin{document}$ {\mathcal O}(h^2) $\end{document}</tex-math></inline-formula>, while <inline-formula><tex-math id="M4">\begin{document}$ {\mathcal O}(h^4) $\end{document}</tex-math></inline-formula> for quadratic basis functions with <inline-formula><tex-math id="M5">\begin{document}$ h $\end{document}</tex-math></inline-formula> a small mesh size. This accuracy can be achieved for any <inline-formula><tex-math id="M6">\begin{document}$ \alpha \in (0, 2) $\end{document}</tex-math></inline-formula> and can be further increased if higher-degree basis functions are chosen. Numerical experiments are provided to approximate the fractional Laplacian and solve the fractional Poisson problems. It shows that if the solution of fractional Poisson problem satisfies <inline-formula><tex-math id="M7">\begin{document}$ u \in C^{m, l}(\bar{ \Omega}) $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M8">\begin{document}$ m \in {\mathbb N} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ 0 < l < 1 $\end{document}</tex-math></inline-formula>, our method has an accuracy of <inline-formula><tex-math id="M10">\begin{document}$ {\mathcal O}(h^{\min\{m+l, \, 2\}}) $\end{document}</tex-math></inline-formula> for constant and linear basis functions, while <inline-formula><tex-math id="M11">\begin{document}$ {\mathcal O}(h^{\min\{m+l, \, 4\}}) $\end{document}</tex-math></inline-formula> for quadratic basis functions. Additionally, our method can be readily applied to approximate the generalized fractional Laplacians with symmetric kernel function, and numerical study on the tempered fractional Poisson problem demonstrates its efficiency.</p>

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Numerical Approximations for the Tempered Fractional Laplacian: Error Analysis and Applications

Cited by 24 publications

References 23 publications

Bilateral Tempered Fractional Derivatives

Bilateral Tempered Fractional Derivatives

Bilateral Tempered Fractional Derivatives

Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization

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